Syzygies in geometry or topology?

Question

I am interested in knowing about the application of Hilbert's Syzygy Theorem (or, for that matter, of the concept of syzygy itself) in geometry or topology, that is, in the fields that have to do with intuitive space. Is there any theorem regulating the translation of the results of a given decomposition into syzigies in algebra into its counterparts or equivalents in space (topology, geometry)? A little bit of historical illustration on that would be great. Thanks in advance.